convex set radially on a sphere ball

Question

Suppose that $C$ is a convex set such that $0\in C\subset\mathbb{R}^n$ and $$\forall x\in \mathbb{R}^n\backslash\{0\},\exists\{x_k\}_k\subset C\text{ s.t. }\lim_{k\to\infty} \frac{x_k}{\|x_k\|}=\frac{x}{\|x\|}$$ The question is whether it can be concluded that there exists a small enough $\epsilon$ satisfying $$ B_\epsilon \subset C$$ So far I didn't come up with any good idea. The convexity seems to play an important role here since without it the statement is wrong.